I think that such squares should be called "exact" or "semicartesian" (where cartesian square = pb).
They should be viewed as the natural self-dual generalisation of pullback and pushout. They appear whenever one studies categories of relations.

1. In an abelian category, I would prefer "exact", or "Hilton-exact".

Hilton considered such squares (for abelian categories), and proved that an equivalent condition is that this square (of proper morphisms) is "bicommutative" in the category of relations (i.e. it commutes and stays commutative when you reverse two "parallel" arrows - as relations).

Plainly:   bicartesian square  =>  pullback  =>  exact;  and dually.

REFERENCE:
P. Hilton, Correspondences and exact squares, in: Proc. Conf. on Categorical Algebra, La Jolla 1965, Springer, pp. 254-271.

2. Studying more general categories of relations, I considered "semicartesian squares"  (f,g, h,k),  defined - in any category - by the following self-dual property (after being commutative, of course):

 Whenever  (f',g', h,k)  and  (f,g, h',k')  commute, also the external square  (f',g', h',k')  commutes

                            B  
          f'            f        h          h'
  A'             A                D             D'
          g'          g       k           k'
                           C

(add slanting arrows  f': A' --> B,  f: A --> B,   etc).

- Again: bicartesian square  =>  pullback  =>  semicartesian,  and dually.

- If pb's  and/or  po's exist, one can give a lot of equivalent properties; eg:

--  (f,g)  and the pb of  (h,k) have the same po (or the same commutative squares out of them).

- In an abelian category, semicartesian amounts to the previous notion.
- In Set, it characterises again those squares which are bicommutative in Rel.

REFERENCE:
M. Grandis, Symétrisations de categories et factorisations quaternaires, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. 14 sez. 1 (1977), 133-207.

3. A 2-dimensional version of this property (actually a STRUCTURE on 2-cells), was introduced by Guitart, and called "H-exact", if I remember well (H for Hilton)

REFERENCES:

- R. Guitart, Carrés exacts et carrés deductifs, Diagrammes 6 (1981), G1-G17.
- R. Guitart and L. Van den Bril, Calcul des satellites et présentations des bimodules à l'aide des carrés exacts, Cahiers Topologie Géom. Différentielle 24 (1983), no. 3, 299-330.
(and some other papers by the same authors).

Best regards

Marco Grandis